Data.Colour.CIE:lightness from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%
Time: 1.8s
Alternatives: 3
Speedup: 1.0×

Specification

?
\[\begin{array}{l} \\ x \cdot 116 - 16 \end{array} \]
(FPCore (x) :precision binary64 (- (* x 116.0) 16.0))
double code(double x) {
	return (x * 116.0) - 16.0;
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * 116.0d0) - 16.0d0
end function

public static double code(double x) {
	return (x * 116.0) - 16.0;
}

def code(x):
	return (x * 116.0) - 16.0

function code(x)
	return Float64(Float64(x * 116.0) - 16.0)
end

function tmp = code(x)
	tmp = (x * 116.0) - 16.0;
end

code[x_] := N[(N[(x * 116.0), $MachinePrecision] - 16.0), $MachinePrecision]

\begin{array}{l}

\\
x \cdot 116 - 16
\end{array}

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 3 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 100.0% accurate, 1.0× speedup?

\[\begin{array}{l} \\ x \cdot 116 - 16 \end{array} \]
(FPCore (x) :precision binary64 (- (* x 116.0) 16.0))
double code(double x) {
	return (x * 116.0) - 16.0;
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * 116.0d0) - 16.0d0
end function

public static double code(double x) {
	return (x * 116.0) - 16.0;
}

def code(x):
	return (x * 116.0) - 16.0

function code(x)
	return Float64(Float64(x * 116.0) - 16.0)
end

function tmp = code(x)
	tmp = (x * 116.0) - 16.0;
end

code[x_] := N[(N[(x * 116.0), $MachinePrecision] - 16.0), $MachinePrecision]

\begin{array}{l}

\\
x \cdot 116 - 16
\end{array}

Alternative 1: 100.0% accurate, 0.0× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 116, -16\right) \end{array} \]
(FPCore (x) :precision binary64 (fma x 116.0 -16.0))
double code(double x) {
	return fma(x, 116.0, -16.0);
}

function code(x)
	return fma(x, 116.0, -16.0)
end

code[x_] := N[(x * 116.0 + -16.0), $MachinePrecision]

\begin{array}{l}

\\
\mathsf{fma}\left(x, 116, -16\right)
\end{array}
Derivation

  1. Initial program 100.0%

    \[x \cdot 116 - 16 \]
  2. Step-by-step derivation

    1. fmm-def100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 116, -16\right)} \]

    2. metadata-eval100.0%

      \[\leadsto \mathsf{fma}\left(x, 116, \color{blue}{-16}\right) \]

  3. Simplified100.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, 116, -16\right)} \]
  4. Add Preprocessing
  5. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup?

\[\begin{array}{l} \\ x \cdot 116 - 16 \end{array} \]
(FPCore (x) :precision binary64 (- (* x 116.0) 16.0))
double code(double x) {
	return (x * 116.0) - 16.0;
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * 116.0d0) - 16.0d0
end function

public static double code(double x) {
	return (x * 116.0) - 16.0;
}

def code(x):
	return (x * 116.0) - 16.0

function code(x)
	return Float64(Float64(x * 116.0) - 16.0)
end

function tmp = code(x)
	tmp = (x * 116.0) - 16.0;
end

code[x_] := N[(N[(x * 116.0), $MachinePrecision] - 16.0), $MachinePrecision]

\begin{array}{l}

\\
x \cdot 116 - 16
\end{array}
Derivation

  1. Initial program 100.0%

    \[x \cdot 116 - 16 \]
  2. Add Preprocessing
  3. Add Preprocessing

Alternative 3: 50.7% accurate, 5.0× speedup?

\[\begin{array}{l} \\ -16 \end{array} \]
(FPCore (x) :precision binary64 -16.0)
double code(double x) {
	return -16.0;
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = -16.0d0
end function

public static double code(double x) {
	return -16.0;
}

def code(x):
	return -16.0

function code(x)
	return -16.0
end

function tmp = code(x)
	tmp = -16.0;
end

code[x_] := -16.0

\begin{array}{l}

\\
-16
\end{array}
Derivation

  1. Initial program 100.0%

    \[x \cdot 116 - 16 \]
  2. Add Preprocessing

  3. Taylor expanded in x around 0 55.2%

    \[\leadsto \color{blue}{-16} \]
  4. Add Preprocessing

Reproduce

?
herbie shell --seed 2024157 
(FPCore (x)
  :name "Data.Colour.CIE:lightness from colour-2.3.3"
  :precision binary64
  (- (* x 116.0) 16.0))